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%I #8 May 27 2024 15:48:10
%S 1,0,1,0,-1,1,0,-1,-2,1,0,-1,-3,-3,1,0,-1,-4,-6,-4,1,0,-1,-5,-10,-10,
%T -5,1,0,-1,-6,-15,-20,-15,-6,1,0,-1,-7,-21,-35,-35,-21,-7,1,0,-1,-8,
%U -28,-56,-70,-56,-28,-8,1,0,-1,-9,-36,-84,-126,-126,-84,-36,-9,1
%N Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).
%H Y. Alp and E. G. Kocer, <a href="https://doi.org/10.1007/s00025-024-02193-5">Exponential Almost-Riordan Arrays</a>, Results Math 79, 173 (2024). See page 8.
%F T(n,0) = A000007(n); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] (2-exp(x))*x^(k-1).
%e The triangle begins:
%e 1;
%e 0, 1;
%e 0, -1, 1;
%e 0, -1, -2, 1;
%e 0, -1, -3, -3, 1;
%e 0, -1, -4, -6, -4, 1;
%e 0, -1, -5, -10, -10, -5, 1;
%e 0, -1, -6, -15, -20, -15, -6, 1;
%e 0, -1, -7, -21, -35, -35, -21, -7, 1;
%e ...
%t T[n_,0]:=KroneckerDelta[n,0]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[(2-Exp[x])x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten
%Y Cf. A000012 (right diagonal), A024000 (subdiagonal), A122958 (row sums), A153881 (k=1).
%Y Triangle A154926 with 1st column A000007.
%K sign,tabl
%O 0,9
%A _Stefano Spezia_, May 26 2024
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